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Bicriterial Approximation for the Incremental Prize-Collecting Steiner-Tree Problem

Authors Yann Disser , Svenja M. Griesbach , Max Klimm , Annette Lutz

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ACM Subject Classification
  • Theory of computation → Online algorithms
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • incremental optimization
  • competitive analysis
  • prize-collecting Steiner-tree

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Abstract

We consider an incremental variant of the rooted prize-collecting Steiner-tree problem with a growing budget constraint. While no incremental solution exists that simultaneously approximates the optimum for all budgets, we show that a bicriterial (α,μ)-approximation is possible, i.e., a solution that with budget B+α for all B ∈ ℝ_{≥ 0} is a multiplicative μ-approximation compared to the optimum solution with budget B. For the case that the underlying graph is a tree, we present a polynomial-time density-greedy algorithm that computes a (χ,1)-approximation, where χ denotes the eccentricity of the root vertex in the underlying graph, and show that this is best possible. An adaptation of the density-greedy algorithm for general graphs is (γ,2)-competitive where γ is the maximal length of a vertex-disjoint path starting in the root. While this algorithm does not run in polynomial time, it can be adapted to a (γ,3)-competitive algorithm that runs in polynomial time. We further devise a capacity-scaling algorithm that guarantees a (3χ,8)-approximation and, more generally, a ((4𝓁 - 1)χ, (2^{𝓁 + 2})/(2^𝓁 -1))-approximation for every fixed 𝓁 ∈ ℕ.

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Yann Disser, Svenja M. Griesbach, Max Klimm, and Annette Lutz. Bicriterial Approximation for the Incremental Prize-Collecting Steiner-Tree Problem. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 47:1-47:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.47

Author Details

Yann Disser
  • Department of Mathematics, Technische Universität Darmstadt, Germany
Svenja M. Griesbach
  • Institute of Mathematics, Technische Universität Berlin, Germany
Max Klimm
  • Institute of Mathematics, Technische Universität Berlin, Germany
Annette Lutz
  • Department of Mathematics, Technische Universität Darmstadt, Germany

Funding

Supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through subprojects A07 and A09 of CRC/TRR154 and under Germany’s Excellence Strategy - The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project IDs: 390685689).

Acknowledgements

We thank Felix Hommelsheim, Alexander Lindermayr, and Jens Schlöter for fruitful discussions and three anonymous referees for their comments that improved the presentation of the paper.

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